A survey of concepts of independence for imprecise probabilities

Our aim in this paper is to clarify the notion of independence for imprecise probabilities. Suppose that two marginal experiments are each described by an imprecise probability model, i.e., by a convex set of probability distributions or an equivalent model such as upper and lower probabilities or previsions. Then there are several ways to define independence of the two experiments and to construct an imprecise probability model for the joint experiment. We survey and compare six definitions of independence. To clarify the meaning of the definitions and the relationships between them, we give simple examples which involve drawing balls from urns. For each concept of independence, we give a mathematical definition, an intuitive or behavioural interpretation, assumptions under which the definition is justified, and an example of an urn model to which the definition is applicable. Each of the independence concepts we study appears to be useful in some kinds of application. The concepts of strong independence and epistemic independence appear to be the most frequently applicable.